To me, the term refers to perseverance against daunting obstacle, daunting resistance, laziness, self-indulgence, or compelling temptation. Finishing what you start. It used to be called "character strength."
Studying late at night for a Chem exam, resisting a donut, keeping climbing when your legs say "No," doing one more bench rep when your arms say "I can't", saying "Hi" to a pretty girl who is too good for you, giving your exasperating guitar practice one more half hour, bucking up for one more damn intimidating job interview, getting through a pile of paperwork. A doctor once told me that the toughest person he knew was a agoraphobic and social phobic who by sheer willpower forced herself to leave her house and re-enter society despite her terror.
Everybody wants to fight against his own limitations, fears, flaws, and weaknesses, and nobody wants to feel mentally or physically weak.
When I consider mental toughness I think of warriors facing a wall of spears or machine guns, but in our (or my) pampered and decadent American life we often have to go out of our way to seek out character tests and character challenges. We can easily avoid most of them if we wish to, but our life is diminished by it.
I think shame, self-disgust, and self-disappointment are some of the unpleasant consequences of confronting some of our weaknesses and limitations. It is failure, and we know it.
My genius trainer and I were discussing the topic a while ago. I told him that part of his added value to me was lending his mental strength to me. When my arms say "No," and he says "Two more - you can do it" - I do it. Alone, I "couldn't." That's my mental weakness. So we can gain strength from relationship. That's part of why marriage is so valuable. Friends, too.
He said that group exercise training works well not so much because of the competition but because of the combined spirit of effort, the esprit de corps. Of course, that group effect is an essential part of military training too. He says the reason most people fail in fitness programs is for lack of spirit and determination, not muscle.
I suppose good habits of toughness and perseverance can be nurtured from within from practice and from without with support and cheerleading. My mental toughness is not good enough to make me happy with myself, and probably never will be.
Here's a Grit Quiz. Not sure if such personality traits are measurable really, or whether they are even traits. Sometimes I have plenty of grit, sometimes very little.
Related: The Personality Secret to Successful Weight Loss
What's your view on the subject?